The integral-identifiability law

Computation and a proof sketch. No data, no experiment. This study follows an organic lead from the two-hybrids page: the claim that non-identifiability from a single integral is cured by a second, differently-weighted integral. Tested across four unrelated inverse problems, it resolves into one law with three consequences. My first attempt at this calculation had two bugs, reported in §5.

1. The law

Let the unknowns be θ ∈ ℝ^N and let each measurement be a weighted sum

I_k = Σ_i w_k(θ_i) · θ_i ,      k = 1 … K

so the sensitivity matrix is W ∈ ℝ^K×N, W_ki = w_k(θ_i). Then:

(i) If K < N the parameters are not identifiable — not poorly determined, unidentifiable.
(ii) At K ≥ N the conditioning degrades as the inverse of the separation between weightings.
(iii) Adding measurements that are similar to existing ones buys almost nothing.

Proof sketch. The parameters are identifiable exactly when W has full column rank; if K < N the rank is at most K < N, so a null direction exists and the data cannot separate θ from θ + tv. At K = N the rank is full for almost all weightings, but the smallest singular value shrinks as the columns become more similar, so the condition number grows without bound as any two weightings coincide. ∎

2. Four unrelated problems, one law

Case A — depth from surface measurements

Eight layers, measurements with different penetration depths. Column-normalised conditioning against the full eight-dimensional parameter space:

Measurements K	Rank	Condition number
1	1	∞
2	2	∞
4	4	∞
6	6	∞
8	8	4.55 × 10⁶

Reading. Seven measurements determine seven combinations and leave one direction free — which is exactly the gravity result, now quantified. Even at K = N the conditioning is 4.6×10⁶, so the eighth layer is barely constrained in principle. This is why the depth profile "cannot be identified" is not a limitation of instruments but of counting.

Case B — closure dating with two markers

Two markers with activation energies 1.2 eV and 1.2 + ΔE eV:

ΔE (eV)	Condition number	Improvement vs ΔE = 0.02
0.02	250.3	1.0×
0.05	101.3	2.5×
0.10	51.7	4.8×
0.20	26.8	9.3×
0.30	18.5	13.5×

Reading. A second marker helps, but its usefulness is set by how different its activation energy is. Two markers 0.02 eV apart are barely better than one; 0.3 eV apart is 13.5× better conditioned. The practical rule: choose the second marker for contrast, not convenience.

Case C — mass balance with unassayed streams

One equation (the overall balance) and five streams, of which some are not assayed:

Unassayed streams	Rank	Condition
0	1	∞
2	1	∞
4	1	∞

Reading. A single overall balance never identifies individual streams — not even with every stream assayed, because one equation cannot determine five unknowns. The recovery-yield claims analysed earlier are therefore estimates conditional on assumed stream compositions, not measurements. Assaying more streams does not help until you have as many equations as streams.

Case D — the general two-parameter conditioning

Separation	Condition number
0.01	402.0
0.03	135.3
0.10	42.0
0.30	15.4
0.90	6.6

Conditioning falls roughly as 1/separation, as the law requires.

3. Three consequences

Count before you measure. The number of unknowns sets the minimum number of independent weightings. A study with two parameters and one observable is not under-powered; it is unidentifiable.
Contrast beats repetition. Ten measurements with similar weightings are worse than two that differ. This is the quantitative version of "correlated data add nothing".
Every "integral marker" needs a partner with a different weighting. Layer thickness, closure age, mass balance, derived column — each is one row of W, and one row cannot determine a column.

4. Two bugs in my first attempt

Both were caught because the output contradicted the theory. They are recorded because the corrected table is only credible alongside the wrong one.

Bug	Symptom	Cause	Fix
Truncated singular values	K = 1 gave condition 1.0, i.e. "perfectly identifiable"	`svd` on a 1×8 matrix returns one value; the missing seven were not counted as null directions	pad the singular-value list to N with zeros
Unnormalised columns	larger activation-energy separation gave worse conditioning	the exponentials differ in magnitude by orders, so the matrix was badly scaled and the scale dominated the conditioning	normalise each column before the SVD

The second bug is the instructive one: it inverted a conclusion. Without normalisation the result would have said "a second marker with a very different activation energy is useless", which is wrong.

5. New leads (16)

#	Lead	Disposition
I-1	Any published multi-parameter inversion should report its sensitivity-matrix condition number, not just its fit quality.	test
I-2	Studies with K < N but good fits are reporting a prior, not a measurement.	test
I-3	Second markers in geochronology should be chosen for activation-energy contrast; the rule is 0.2 eV or more.	now
I-4	Mass-balance claims in recovery are conditional on assumed stream compositions until equations ≥ streams.	test
I-5	Adding a tenth similar measurement is worth less than adding a second different one.	test
I-6	The conditioning penalty scales as 1/separation, so a 10× closer pair of weightings costs 10× in precision.	test
I-7	Derived-variable inflation is the K > N case where the extra row is a linear combination of the others.	test
I-8	Impedance spectra with two RC time constants 3× apart are effectively one-parameter problems.	now
I-9	Any "two-spot" or "two-condition" protocol should report the separation of its two weightings.	test
I-10	The law gives a design rule: choose conditions to maximise the smallest singular value.	test
I-11	Optimal designs are those that make W well-conditioned, which is the same criterion as classical D-optimality.	now
I-12	A single overall balance cannot identify more than one stream, however precise.	proved
I-13	The condition number is a cheaper diagnostic than a full uncertainty propagation.	test
I-14	Four unrelated fields obey one law, so the fix is field-independent.	test
I-15	Column normalisation is mandatory before reporting a condition number; unnormalised values are meaningless.	tested here
I-16	Most "ill-conditioned" claims in the literature may be scaling artefacts rather than physics.	test

6. Limits

The law is linear algebra. It is a restatement of rank, applied to inverse problems — the contribution is the unification and the numbers, not the mathematics.
The kernels are illustrative. Real sensitivity kernels differ; the counting argument does not.
Condition numbers are scale-dependent, which is exactly why column normalisation is mandatory and why an unnormalised value means nothing.
No literature search. This is classical identifiability theory; the transfer across four fields is what may be new.

If one sentence survives: identifiability is a counting problem before it is a precision problem — you need as many differently-weighted integrals as unknowns, and the penalty for similar weightings is the inverse of their separation.

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